**
Andrei
Teleman,
Aix-Marseille
University**

**
"Moduli
spaces
of
holomorphic
bundles
framed
along
a
real
hypersurface"**

Let
\(X\)
be
a
connected,
compact
complex
manifold,
and
\(S\subset
X\)
be
a
separating
real
hypersurface.
\(X\)
decomposes
as
a
union
of
compact
complex
manifolds
with
boundary
\(\bar
X^\pm\)
with
\(\bar
X^+\cap
\bar
X^-=S\).
Let
\(\mathcal{M}\)
be
the
moduli
space
of
\(S\)-framed
holomorphic
bundles
on
\(X\),
i.e.
of
pairs
\((E,\theta)\)
(of
fixed
topological
type)
consisting
of
a
*
holomorphic*
bundle
\(E\)
on
\(X\)
endowed
with
a
*
differentiable*
trivialization
\(\theta\)
on
\(S\).
This
moduli
space
is
the
main
object
of
a
joint
research
project
with
Matei
Toma.

The problem addressed in my talk: compare, via the obvious restriction maps, the moduli space \(\mathcal{M}\) with the corresponding Donaldson moduli spaces \(\mathcal{M}^\pm\) of boundary framed holomorphic bundles on \(\bar X^\pm\). The restrictions to \(\bar X^\pm\) of an \(S\)-framed holomorphic bundle \((E,\theta)\) are boundary framed formally holomorphic bundles \((E^\pm,\theta^\pm)\) which induce, via \(\theta^\pm\), the same tangential Cauchy-Riemann operators on the trivial bundle on \(S\). Therefore one obtains a natural map from \(\mathcal{M}\) into the fiber product \(\mathcal{M}^-\times_\mathcal{C}\mathcal{M}^+\) over the space \(\mathcal{C}\) of Cauchy-Riemann operators on the trivial bundle on \(S\).

Our
result
states:
*
this
map
is
bijective.*
Note
that,
by
theorems
due
to
S.
Donaldson
and
Z.
Xi,
the
moduli
spaces
\(\mathcal{M}^\pm\)
can
be
identified
with
moduli
spaces
of
boundary
framed
Hermitian
Yang-Mills
connections.

This seminar will be held both online and in person:

- Room: MC 5417
- Zoom link: https://uwaterloo.zoom.us/j/96883292635?pwd=KytGYnEvRmhyTTV1NC9Gc2dnT05oQT09